Oscillating Slingshot

A ball of mass m is connected to two rubber bands of length, L, each under tension T, as in Figure P12.49. The ball is displaced by a small distance y perpendicular to the length of the rubber bands.

(a) Assuming that the tension does not change, show that the restoring force is -2yT/L .
(b) Assuming that the tension does not change, show that the system exhibits simple harmonic motion with an angular frequency .

I have no idea how to solve this. I think it has something to do with F=ma, but I don't know exactly what.

Let's call the direction in which the displacement has been made the positive y-axis, and let the initial length of the bands lie along the x-axis.

After the displacement y, the rubber bands each make an angle theta with the x-axis.

Now, resolve the components of each of the tensions T along the y-axis and the x-axis. The horizontal tensions are each Tcos(theta) and balance each other. The y-component of each of the tension T is Tsin(theta). So, the force acting on the particle is 2Tsin(theta) towards the origin. Then we can write,

Part a is fairly simple, if you look at the vector diagram, you can see that the downward component of the tension force from EACH rubber band is given by -Tsin(theta), where sin(theta) = y/L (which should be clear from the geometry in the figure.

Then, in the vertical direction, Newton's second law becomes:

may = -2Ty/L

Shooting star is using calculus to write this as:

[tex] m \frac{d^2 y}{dt^2} = -2T\frac{y}{L} [/tex]

Where the symbol [tex] \frac{d^2}{dt^2} [/tex] should be taken to be one whole symbol that is being applied to y, and it means "take the second dervative of y with respect to time." If you don't know what that means, don't worry about it. This is the differential equation of a simple harmonic oscillator, but I don't think your teacher intended for you to use differential equations to solve for it. So, given that you have a RESTORING FORCE i.e. our equation is of the general form:

may = -ky

where k is some constant, it seems pretty clear that we'll get oscillation. What rules has your teacher taught you about deducing the frequency of oscillation (omega) using this type of equation as a starting point?